Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

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Is the space of Radon measures a Polish space or at least separable?

Background: I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier ...
Mark's user avatar
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11 votes
2 answers
672 views

Character theory and Quantum Chemistry

Who (presumably a chemist) realized first the efficiency of character theory in calculations of orbitals of atoms? In which year?
Denis Serre's user avatar
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6 votes
0 answers
128 views

Generalization of pseudogroups

Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between manifolds,...
Joshua Meyers's user avatar
5 votes
2 answers
695 views

What is this disintegration-like theorem?

This is cross-posted at MSE. I'm looking for a reference for the following result. It seems like it must be known, or follow quickly from something known, but I have not been able to find it in any ...
user435571's user avatar
7 votes
1 answer
482 views

Furthest distance half the diameter?

Let $S$ be the surface of a convex body, polyhedral or smooth, embedded in $\mathbb{R}^3$. For a point $x \in S$, let $F(x)$ be the set of furthest points from $x$, measured by shortest paths on the ...
Joseph O'Rourke's user avatar
4 votes
0 answers
247 views

p cohomological dimension of a profinite group

I would like to know what is the $p$-cohomological dimension of $\textrm{Gal}(\mathbb{Q}_S/\mathbb{Q}_{cyc})$. Here $S$ is a finite set of primes containing $p$ and the Archimedean primes and $\mathbb{...
debanjana's user avatar
  • 1,161
1 vote
0 answers
154 views

Mixed Hodge structures over $F\otimes \mathbb{R}$

Let $F$ be a number field. Nekovàř, on page 18 of Values of L-functions and p-adic cohomology, is referring to the category of mixed Hodge structures over $F\otimes_{\mathbb{Q}} \mathbb{R}$. Can ...
Stabilo's user avatar
  • 1,479
6 votes
1 answer
212 views

Number of irreducible representations of $SO_3(\mathfrak{o}/\mathfrak{p}^l)$

$\DeclareMathOperator\SO{SO}$Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $\mathfrak{o}$ denote the ring of integers, with maximal ideal $\mathfrak{p}$. Let $G_l$ denote the finite group $\...
nikola karabatic's user avatar
3 votes
0 answers
50 views

exterior problem for fractional Laplacian

Does there exist a theory of fractional laplacian on exterior domains such as $$ \ \ \left\{\begin{aligned} (-\Delta)^{s} u&= 0 &&\text{in } \mathbb R^N\setminus \mathbb B \\ u & ...
GabS's user avatar
  • 407
6 votes
1 answer
265 views

Borel / Wadge hierarchies on subsets closed under prepending a finite prefix

I'm interested in subsets $X$ of the Cantor space ($2^\omega$) or the Baire space ($\omega^\omega$) that are closed under prepending an arbitrary finite prefix: $$ (x_1, x_2, \dots) \in X \implies (...
user1020406's user avatar
4 votes
1 answer
316 views

What about $n^{\frac{1}{x}+\frac{1}{y}}+n^{\frac{1}{y}+\frac{1}{z}}=n^{\frac{1}{z}+\frac{1}{x}}$ over positive integers?

I've considered the following equation for positive integers $x,y,z\geq 1$, and for positive integers $n\geq 2$ $$n^{\frac{1}{x}+\frac{1}{y}}+n^{\frac{1}{y}+\frac{1}{z}}=n^{\frac{1}{z}+\frac{1}{x}},\...
user142929's user avatar
4 votes
1 answer
1k views

Does the Legendre-Fenchel transform/convex conjugate of strongly convex functions have any desirable properties?

It is well known in convex analysis that when a closed, proper, function $f$ is Legendre-type, that is, essentially strictly convex and essentially smooth, the Legendre transform yields a dual ...
Concu Bine's user avatar
17 votes
4 answers
2k views

Differential geometry applied to biology

This was originally a question posted here on MathSE. But I'll ask again here to see if I can get some different answers. I'm looking for current areas of research which apply techniques from ...
Argent's user avatar
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0 votes
0 answers
129 views

What about an alternative formulation for different prime constellations in the spirit of Suzuki's theorem for twin primes?

It is known that the twin prime conjecture is a special case of the $k$-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld. On the ...
user142929's user avatar
6 votes
2 answers
404 views

Reduction to Lie algebra version of fundamental lemma?

Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration. For the purposes of the trace formula, one actually needs the fundamental ...
Spencer Leslie's user avatar
8 votes
0 answers
262 views

A diagram in the proof of Theorem 2.5.5 of 'Cohomology of Number Fields' and the Tate Spectral Sequence

I've been reading the book 'Cohomology of Number Fields' for years. But I couldn't check the commutativity of the diagram on page 126 until now. So I ask for help. The diagram is induced by taking ...
gualterio's user avatar
  • 1,043
7 votes
0 answers
124 views

Removing rows to reduce the rank

What is the smallest number of rows one can delete from a matrix to reduce its rank (by $1$)? Is there any standard name / notation for this characteristic? Has it been studied? I am in fact ...
Seva's user avatar
  • 22.8k
7 votes
2 answers
384 views

Is every metric uniformly close to a metric with negative scalar curvature?

Let $M$ be a smooth manifold with non-empty boundary. Let $g$ be a smooth Riemannian metric on $M$. Is the following true? For every $\epsilon >0$ there exist a Riemannian metric $g_{\epsilon}$ ...
Asaf Shachar's user avatar
  • 6,611
4 votes
1 answer
173 views

Intrinsic volumes of non-polyconvex, non-compact sets

I am reposting this question I asked and bountied on Math SE, which has been upvoted but not answered or commented on. The intrinsic volumes (AKA Minkowski Functionals or, with different ...
Joe Previdi's user avatar
13 votes
3 answers
798 views

Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function?

For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum_{k=1}^n n\text{ mod }k,\tag{1}$$ the sum of remainders function, the arithmetic function A004125 from the OEIS. Example. We'...
user142929's user avatar
4 votes
1 answer
193 views

Reference for definition of residue of a differential form, in all characteristics

What is the standard reference for a definition , valid in all characteristics, of the residue in a point of a rational differential form on a curve?
IMeasy's user avatar
  • 3,717
4 votes
0 answers
937 views

Next step in studying arithmetic geometry

This relates to this post. I want to study arithmetic, such as Fermat's last theorem, Faltings' theorem, Mazur's torsion points theorem, Weil conjecture and so on. For understanding these theorems (...
k.j.'s user avatar
  • 1,352
9 votes
0 answers
788 views

How many ways are there to teach class field theory?

I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now. I definitely ...
Mikhail Bondarko's user avatar
3 votes
0 answers
150 views

A variant on the Higman-Thompson groups

Let $C = \mathbb{Z}/d\mathbb{Z}$ ($d \ge 0$). Let $D = \langle a_c : c \in C, t \mid a^2_c = t^d = 1, ta_ct^{-1} = a_{c+1} \rangle$. let $E$ be the subgroup generated by $\{a_c : c \in C\}$ and let $...
Colin Reid's user avatar
  • 4,678
1 vote
1 answer
114 views

Reference requence: scheme of complete homomorphisms of rank $r$ via blowups

I'm reading these notes where it states in section $3$: (transcribed because I can't post image) Step 1. Introduce the stacks of degenerated and iterated shtukas which extends that of shtukas. This ...
edgarlorp's user avatar
  • 113
0 votes
1 answer
367 views

Reference request: Oldest books on analytic geometry with unsolved exercises?

Per the title, what are some of the oldest books on analytic geometry out there with unsolved exercises? Maybe there are some hidden gems from before the 20th century out there.
Squid with Black Bean Sauce's user avatar
7 votes
0 answers
213 views

Reference request: retracts are summand inclusions in additive $\infty$-categories

Suppose that $\mathcal{A}$ is an additive $\infty$-category. By this I mean that $\mathcal{A}$ is pointed, semi-additive (i.e., admits biproducts, which I will call direct sums and denote by $\oplus$),...
Yonatan Harpaz's user avatar
6 votes
0 answers
123 views

Countable-to-one factors of measure preserving systems do not change entropy

It is known that if $\psi$ is a factor map between probability measure preserving systems $(X,\mathscr{X},\mu,T)$ and $(Y,\mathscr{Y},\nu,S)$ is countable-to-one almost everywhere, then $h(\mu,T)=h(\...
Dominik Kwietniak's user avatar
0 votes
1 answer
324 views

Inclusion of closed submanifolds of a manifold

Consider a smooth compact manifold $M$ of dimension $n$, with or without boundary. Choose a submanifold $N$ of $M$ of dimension $k$, where $1 \leq k \leq n - 1$, such that $N$ is either without ...
SMS's user avatar
  • 1,293
1 vote
1 answer
121 views

Reference request concerning order statistics from the uniform distribution

Let $U_1,\dots,U_n$ be iid random variables uniformly distributed on the interval $[0,1]$, with the corresponding order statistics $U_{(1)}\le\dots\le U_{(n)}$. Let $G_i:=U_{(i+1)}-U_{(i)}$ for $i=0,\...
Iosif Pinelis's user avatar
1 vote
0 answers
86 views

Local coefficient systems in cohomology

Let $\mathcal F$ be a locally constant sheaf with values in $\mathbb C$ on a nice enough space, say a compact manifold. The etale space of $\mathcal F$ defines a covering $p: \tilde X \to X$. Is ...
trabs's user avatar
  • 11
13 votes
0 answers
1k views

Is there a slick proof of the fundamental theorem of dimension theory?

The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of ...
display llvll's user avatar
20 votes
2 answers
2k views

Applications of number theory in dynamical systems

I am looking for references (or ways to find references) on significant and/or recent applications of techniques in number theory to problems in the areas of dynamical systems and nonlinear dynamics. ...
J W's user avatar
  • 748
2 votes
0 answers
129 views

Weak Lefschetz property Jacobian ring smooth hypersurface

Let $A_{.}$ be a graded commutative ring. We say that $A_{.}$ satisfies the weak Lefschetz property if for generic $L \in A_1$ the multiplication maps $ \times L : A_i \longrightarrow A_{i+1}$ has ...
Libli's user avatar
  • 7,210
2 votes
0 answers
85 views

State-of-the-Art algorithms for bilevel optimization

I want to numerically solve a bilevel optimization problem of the form $$ \min_y f(y, \hat x(y)), \qquad \hat x(y) = \arg\min_x g(x, y) $$ (for simplicity assume that $\min_x g(x, y)$ exists and is ...
Hyperplane's user avatar
12 votes
2 answers
962 views

Higman's lemma and a manuscript of Erdős and Rado

Motivated by a problem in factorization theory, I've recently proved the following: Theorem. If $X$ is a non-empty finite alphabet and $\mathcal W$ an infinite subset of the free semigroup, $X^\ast$...
Salvo Tringali's user avatar
5 votes
1 answer
214 views

Homologous quotient of fundamental groupoid

Let $X$ be a connected space and $\Pi_1(X)$ be its fundamental groupoid. We consider the homologous relation $\mathcal R$ on every morphism space: $f,g\in \Pi_1(X)(p,q)$ are related if the singular ...
Hang's user avatar
  • 2,719
3 votes
1 answer
582 views

Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds

On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
user144878's user avatar
9 votes
1 answer
294 views

An extrapolation method

I've stumbled upon a method of extrapolation that I haven't seen before. We are trying to approximate $f(0)$ for a certain function $f$, which we have only measured at points $x_0, \ldots, x_N$ in ...
Robert Israel's user avatar
3 votes
0 answers
143 views

Upper bound on the geodesic distance in a Lipschitz domain

I was wondering if the following result is true. If yes, could you please suggest a reference. The result seems to have been used at several papers without quoting any reference. Is the proof ...
Tatin's user avatar
  • 895
-2 votes
1 answer
255 views

Is the conjecture $min(A,B) \le rad(ABC)$ new and correct? [closed]

$\DeclareMathOperator\rad{rad}$Conjecture: If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$. If the ...
Đào Thanh Oai's user avatar
2 votes
2 answers
287 views

Reference request on computational schemes for $\inf_{x\in\Omega^n}\sup_{y\in\mathbb R^n}F(x,y)$

Let $\Omega\subset \mathbb R^d$ be compact, $\rho$ be a density function on $\Omega$ and $p_1,\ldots, p_n\in (0,1)$ be weights satisfying $\int_{\Omega}\rho(z)dz=1=\sum_{k=1}^n p_k$. We consider the ...
user avatar
2 votes
0 answers
162 views

vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves

I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are: $H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
Somatic Custard's user avatar
0 votes
0 answers
23 views

A linear map satisfying the given property

Let $A$ and $B$ be two Banach algebras such that $B$ is a Banach $A$-bimodue and $T:A\rightarrow B$ a linear map satisfying $T(aa')=aT(a')+T(a)a'+T(a)T(a')$ for all $a,a'\in A$. If the algerba ...
Fermat's user avatar
  • 167
3 votes
0 answers
125 views

$\left< 15\right>^7/15$-womcode construction

In the article Womcodes constructed with projective geometries Frans Merkx constructed several good wom-codes (write-once memory codes, see How to reuse a "write-once" memory by Rivest & Shamir ...
Alexey Ustinov's user avatar
2 votes
2 answers
247 views

Reference request on Min-Max theorem

Consider the following min-max problem $$\inf_{x\in M} \sup_{y\in N} F(x,y),$$ where $F: M\times N\to\mathbb R$ is Lipschitz and $y\mapsto F(x,y)$ is concave for all $x\in M$. Could we derive $\...
user avatar
3 votes
0 answers
153 views

Using the Hilbert symbol to find nice field extensions

Let $p$ and $q$ be (not necessarily distinct) odd primes and let $F=\mathbb{Q}_p(\mu_q)$. The $q^{th}$ Hilbert symbol induces a non-degenerate alternating form $$(\cdot,\cdot)_q:F^\times/(F^\times)^q\...
Spencer Leslie's user avatar
6 votes
0 answers
189 views

Which ring spectra are homotopy limits of simpler ones?

Most surely I will tag this by reference request: I am sure very much is known about this question, I am just too ignorant to even guess where to look. What makes me feel especially foolish is the ...
მამუკა ჯიბლაძე's user avatar
2 votes
0 answers
130 views

Hypersurfaces whose unit normal $N$ satisfies $[N,X] =0$ for every tangent vector field $X$

Let $M$ be a hypersurface of a Riemannian manifold, and assume that $M$ satisfies the following property: For each $p \in M$, given a unit normal vector field $N$ defined in a neighborhood $U$ of $...
Matteo Raffaelli's user avatar
3 votes
1 answer
168 views

Translation to English of Brillouin's analysis of Airy's integral

I am trying to read the following paper by Leon Brillouin (the part on page 16 onwards): Léon Brillouin, Sur une méthode de calcul approchée de certaines intégrales dite méthode du col, Annales ...
Alan's user avatar
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